
The hypothesis that \(d\ge 3\)  cannot be weakened to \(d\ge 2\) : setting \(d=2\)  and \(c_1=1\) , \(c_2=-1\) , the set \(\vec{c}\cdot A\)  is simply \(A-A\) . An example of a set \(A\subset \mathbb {Z}\)  having \(d^*(A)>0\)  such that \(A-A\)  does not contain a Bohr neighborhood of 0 is provided by the main result of [1]} - see [2]} for an explanation of why [1]} implies this. Such an example in \(\mathbb {F}_2^\omega \)  is constructed in [4]}, and in \(\mathbb {F}_p^\omega \)  for odd primes \(p\)  in [5]}. In fact there is an \(A\subset \mathbb {Z}\)  with \(d^*(A)>0\)  where \(A-A\)  does not contain a Bohr neighborhood of any \(n\in \mathbb {Z}\) , as proved in [6]}.

The hypothesis that \(c_iG\)  has finite index in \(G\)  cannot be omitted. To see this, we use the fact that for each prime \(p\)  and \(\varepsilon >0\) , the group \(G=\mathbb {F}_p^{\omega }\)  contains a set \(A\)  having \(d^*(A)>\frac{1}{2}-\frac{1}{2p}-\varepsilon \)  such that \(A-A\)  does not contain a Bohr neighborhood of any element of \(G\)  (see [5]}). Setting \(c_1=p\) , \(c_2=1\)  and \(c_3=-1\) , we have \(c_1+c_2+c_3=p\) , but \(c\cdot A \subseteq pA+A-A =A-A\) .

The hypothesis that \(q|\sum c_i\)  cannot be omitted. For example, in \(\mathbb {Z}\) , the set \(A\)  of odd integers has \(d^*(A)=1/2\) , and \(A-A+A=A\) , which does not contain a Bohr neighborhood of 0. Nevertheless, \(A-A+A\)  contains Bohr neighborhoods of many elements of \(A\)  whenever \(d^*(A)>0\) ; see [2]}, [9]}.


